Group algebras

This functionality has been moved to sage.categories.algebra_functor.

sage.algebras.group_algebra.GroupAlgebra(G, R=Integer Ring)

Return the group algebra of \(G\) over \(R\).

INPUT:

• \(G\) – a group

• \(R\) – (default: \(\ZZ\)) a ring

EXAMPLES:

The group algebra \(A=RG\) is the space of formal linear combinations of elements of \(G\) with coefficients in \(R\):

Sage

sage: G = DihedralGroup(3)
sage: R = QQ
sage: A = GroupAlgebra(G, R); A
Algebra of Dihedral group of order 6 as a permutation group over Rational Field
sage: a = A.an_element(); a
() + (1,2) + 3*(1,2,3) + 2*(1,3,2)

Python

>>> from sage.all import *
>>> G = DihedralGroup(Integer(3))
>>> R = QQ
>>> A = GroupAlgebra(G, R); A
Algebra of Dihedral group of order 6 as a permutation group over Rational Field
>>> a = A.an_element(); a
() + (1,2) + 3*(1,2,3) + 2*(1,3,2)

This space is endowed with an algebra structure, obtained by extending by bilinearity the multiplication of \(G\) to a multiplication on \(RG\):

Sage

sage: A in Algebras
True
sage: a * a
14*() + 5*(2,3) + 2*(1,2) + 10*(1,2,3) + 13*(1,3,2) + 5*(1,3)

Python

>>> from sage.all import *
>>> A in Algebras
True
>>> a * a
14*() + 5*(2,3) + 2*(1,2) + 10*(1,2,3) + 13*(1,3,2) + 5*(1,3)

GroupAlgebra() is just a short hand for a more general construction that covers, e.g., monoid algebras, additive group algebras and so on:

Sage

sage: G.algebra(QQ)
Algebra of Dihedral group of order 6 as a permutation group over Rational Field

sage: GroupAlgebra(G,QQ) is G.algebra(QQ)
True

sage: M = Monoids().example(); M
An example of a monoid:
the free monoid generated by ('a', 'b', 'c', 'd')
sage: M.algebra(QQ)
Algebra of An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd')
        over Rational Field

Python

>>> from sage.all import *
>>> G.algebra(QQ)
Algebra of Dihedral group of order 6 as a permutation group over Rational Field

>>> GroupAlgebra(G,QQ) is G.algebra(QQ)
True

>>> M = Monoids().example(); M
An example of a monoid:
the free monoid generated by ('a', 'b', 'c', 'd')
>>> M.algebra(QQ)
Algebra of An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd')
        over Rational Field

See the documentation of sage.categories.algebra_functor for details.

class sage.algebras.group_algebra.GroupAlgebra_class(R, basis_keys=None, element_class=None, category=None, prefix=None, names=None, **kwds)

Bases: CombinatorialFreeModule